Vortex dynamics for the Gross-Pitaevskii equation

TL;DR

This paper rigorously derives the asymptotic behavior of vortex dynamics in the Gross-Pitaevskii equation, showing that at leading order it follows the classical Helmholtz-Kirchhoff system, with corrections described by a linear wave equation.

Contribution

It provides a rigorous derivation of vortex dynamics asymptotics for the Gross-Pitaevskii equation, including explicit vortex solutions and correction terms.

Findings

Vortex solutions with degree ±1 are constructed for any number of vortices.

Vortex dynamics are governed by the Helmholtz-Kirchhoff system at leading order.

First correction to the dynamics is described by a linear wave equation.

Abstract

We rigorously establish the formal asymptotics of Neu for Gross-Pitaevskii vortex dynamics in the plane. Given any integer $n\geq2$, we construct a family of $n$-vortex solutions with vortices of degree $\pm1$, and describe precisely the solution profile and associated vortex dynamics on an arbitrarily large, finite time interval. We compute an asymptotic expansion of the vortex positions in terms of the vortex core size $\epsilon>0$, and show that the dynamics is governed at leading order as $\epsilon\to0$ by the classical Helmholtz-Kirchhoff system. Moreover, we show that the first correction to the leading order dynamics is determined by the solution of a linear wave equation, justifying a formal expansion found by Ovchinnikov and Sigal.